Sorting and selection – Part 2 Prof. Noah Snavely CS1114

2 Administrivia  Assignment 1 due tomorrow by 5pm  Assignment 2 will be out tomorrow –Two parts: smaller part due next Friday, larger part due in two weeks  Quiz 2 next Thursday 2/12 –Coverage through next Tuesday (topics include running time, sorting) –Closed book / closed note

Recap from last time: sorting  If we sort an array, we can find the k th largest element in constant (O(1)) time –For all k, even for the median (k = n/2)  Sorting algorithm 1: Selection sort –Running time: O(n 2 )  Sorting algorithm 2: Quicksort –Running time: O(?) 3 ?

Quicksort 1. Pick an element (pivot) 2. Compare every element to the pivot and partition the array into elements pivot 3. Quicksort these smaller arrays separately 4

Quicksort: worst case  With a bad pivot this algorithm does quite poorly –Degrades to selection sort –Number of comparisons will be O(n 2 )  The worst case occurs when the array is already sorted –We could choose the average element instead of the first element 5

6 Quicksort: best case  With a good choice of pivot the algorithm does quite well  What is the best possible case? –Selecting the median  How many comparisons will we do? –Every time quicksort is called, we have to: % Compare all elements to the pivot

How many comparisons? (best case)  Suppose length(A) == n  Round 1: Compare n elements to the pivot … now break the array in half, quicksort the two halves …  Round 2: For each half, compare n / 2 elements to each pivot (total # comparisons = n) … now break each half into halves …  Round 3: For each quarter, compare n / 4 elements to each pivot (total # comparisons = n) 7

How many comparisons? (best case) How many rounds will this run for? …

How many comparisons? (best case)  During each round, we do a total of n comparisons  There are log n rounds  The total number of comparisons is n log n  In the best case quicksort is O(n log n) 9

Can we expect to be lucky?  Performance depends on the input  “Unlucky pivots” (worst-case) give O(n 2 ) performance  “Lucky pivots” give O(n log n) performance  For random inputs we get “lucky enough” – expected runtime on a random array is O(n log n)  Can we do better? 10

11 Back to the selection problem  Can solve with sorting  Is there a better way?  Rev. Charles L. Dodgson’s problem –Based on how to run a tennis tournament –Specifically, how to award 2 nd prize fairly

12 How many teams were in the tournament? How many games were played? Which is the second-best team?

Finding the second best team  Could use quicksort to sort the teams  Step 1: Choose one team as a pivot (say, Arizona)  Step 2: Arizona plays every team  Step 3: Put all teams worse than Arizona in Group 1, all teams better than Arizona in Group 2 (no ties allowed)  Step 4: Recurse on Groups 1 and 2  … eventually will rank all the teams … 13

Quicksort Tournament  (Note this is a bit silly – AZ plays 63 games)  This gives us a ranking of all teams –What if we just care about finding the 2 nd -best team? 14 Quicksort Tournament Step 1: Choose one team (say, Arizona) Step 2: Arizona plays every team Step 3: Put all teams worse than Arizona in Group 1, all teams better than Arizona in Group 2 (no ties allowed) Step 4: Recurse on groups 1 and 2 … eventually will rank all the teams … Quicksort Tournament Step 1: Choose one team (say, Arizona) Step 2: Arizona plays every team Step 3: Put all teams worse than Arizona in Group 1, all teams better than Arizona in Group 2 (no ties allowed) Step 4: Recurse on groups 1 and 2 … eventually will rank all the teams …

Modifying quicksort to select  Suppose Arizona beats 36 teams, and loses to 27 teams  If we just want to know the 2 nd -best team, how can we save time? 15 < 27 teams36 teams < {{ Group 2Group 1

Modifying quicksort to select – Finding the 2 nd best team 16 < 27 teams36 teams < {{ Group 2Group 1 < 10 teams16 teams < { Group 2.2 { Group 2.1 < 2 teams 7 teams <

Modifying quicksort to select – Finding the 32 nd best team 17 < 27 teams36 teams < {{ Group 2Group 1 < 15 teams20 teams < { Group 1.2 { Group Q: Which group do we visit next? - The 32 nd best team overall is the 4 th best team in Group 1

18 Find k th largest element in A (< than k-1 others) MODIFIED QUICKSORT(A, k):  Pick an element in A as the pivot, call it x  Divide A into A1 ( x)  If k < length(A3) –MODIFIED QUICKSORT (A3, k)  If k > length(A2) + length(A3) –Let j = k – [length(A2) + length(A3)] –MODIFIED QUICKSORT (A1, j)  Otherwise, return x A = [ ]

Modified quicksort  We’ll call this quickselect  Let’s consider the running time… 19 MODIFIED QUICKSORT(A, k):  Pick an element in A as the pivot, call it x  Divide A into A1 ( x)  If k < length(A3) –Find the element < k others in A3  If k > length(A2) + length(A3) –Let j = k – [length(A2) + length(A3)] –Find the element < j others in A1  Otherwise, return x MODIFIED QUICKSORT(A, k):  Pick an element in A as the pivot, call it x  Divide A into A1 ( x)  If k < length(A3) –Find the element < k others in A3  If k > length(A2) + length(A3) –Let j = k – [length(A2) + length(A3)] –Find the element < j others in A1  Otherwise, return x

20 What is the running time of:  Finding the 1 st element? –O(1) (effort doesn’t depend on input)  Finding the biggest element? –O(n) (constant work per input element)  Finding the median by repeatedly finding and removing the biggest element? –O(n 2 ) (linear work per input element)  Finding the median using quickselect? –Worst case? O(n^2) –Best case? O(n)

Quickselect – “medium” case  Suppose we split the array in half each time (i.e., happen to choose the median as the pivot)  How many comparisons will there be? 21

How many comparisons? (“medium” case)  Suppose length(A) == n  Round 1: Compare n elements to the pivot … now break the array in half, quickselect one half …  Round 2: For remaining half, compare n / 2 elements to the pivot (total # comparisons = n / 2) … now break the half in half …  Round 3: For remaining quarter, compare n / 4 elements to the pivot (total # comparisons = n / 4) 22

Number of comparisons = n + n / 2 + n / 4 + n / 8 + … + 1 = ?  The “medium” case is O(n)! 23 How many comparisons? (“medium” case)

24 Quickselect  For random input this method actually runs in linear time (beyond the scope of this class)  The worst case is still bad  Quickselect gives us a way to find the k th element without actually sorting the array!

Quickselect  It’s possible to select in guaranteed linear time (1973) –Rev. Dodgson’s problem –But the code is a little messy And the analysis is messier  Beyond the scope of this course 25

26 Back to the lightstick  By using quickselect we can find the 5% largest (or smallest) element –This allows us to efficiently compute the trimmed mean

27 What about the median?  Another way to avoid our bad data points: –Use the median instead of the mean kids, avg. weight= 40 lbs 1 Arnold, weight = 236 lbs Median: 40 lbsMean: (12 x ) / 13 = 55 lbs

Median vector  Mean, like median, was defined in 1D –For a 2D mean we used the centroid –Mean of x coordinates and y coordinates separately Call this the “mean vector” –Does this work for the median also? 28

29 What is the median vector?  In 1900, statisticians wanted to find the “geographical center of the population” to quantify westward shift  Why not the centroid? –Someone being born in San Francisco changes the centroid much more than someone being born in Indiana  What about the “median vector”? –Take the median of the x coordinates and the median of the y coordinates separately

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31 Median vector  A little thought will show you that this doesn’t really make a lot of sense –Nonetheless, it’s a common solution, and we will implement it for CS1114 –In situations like ours it works pretty well  It’s almost never an actual datapoint  It depends upon rotations!

32 Can we do even better?  None of what we described works that well if we have widely scattered red pixels –And we can’t figure out lightstick orientation  Is it possible to do even better? –Yes!  We will focus on: –Finding “blobs” (connected red pixels) –Summarizing the shape of a blob –Computing orientation from this  We’ll need brand new tricks!

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